Anomalous self-similar solutions of exponential type for the subcritical fast diffusion equation with weighted reaction

Abstract

We prove existence and uniqueness of the branch of the so-called \emph{anomalous eternal solutions} in exponential self-similar form for the subcritical fast-diffusion equation with a weighted reaction term $${\partial }_{t}u=\Delta u^m+|x{|}^{\sigma }{u}^p,$$ posed in ${\Re }^N$ with $N\ge 3$, where $$0<m<m_c=\frac{N-2}{N},p>1,$$ and the critical value for the weight $$\sigma =\frac{2(p-1)}{1-m}.$$ The branch of exponential self-similar solutions behaves similarly as the well-established anomalous solutions to the pure fast diffusion equation, but without a finite time extinction or a finite time blow-up, and presenting instead a \emph{change of sign of both self-similar exponents} at $m=m_s=(N-2)/(N+2)$, leading to surprising qualitative differences. In this sense, the reaction term we consider realizes a \emph{perfect equilibrium} in the competition between the fast diffusion and the reaction effects.

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